Mathematical Methods for Physics and Engineering : A Comprehensive Guide

31.7 HYPOTHESIS TESTING

The sum of squares in the denominator of (31.112) may be put into the form

∑

i(xi−μ^0 )

(^2) =N( ̄x−μ 0 ) (^2) +∑
i(xi−x ̄)
(^2).
Thus, on dividing the numerator and denominator in (31.112) by
∑
i(xi− ̄x)
(^2) and
rearranging, the generalised likelihood ratioλcan be written
λ=
(
1+
t^2
N− 1
)−N/ 2
,
where we have defined the new variable
t=
̄x−μ 0
s/
√
N− 1

. (31.113)

Sincet^2 is a monotonically decreasing function ofλ, the corresponding rejection

region ist^2 >c,wherecis a positive constant depending on the required

significance levelα. It is conventional, however, to usetitself as our test statistic,

in which case our rejection region becomes two-tailed and is given by

t<−tcrit and t>tcrit, (31.114)

wheretcritis the positive square root of the constantc.

The definition (31.113) and the rejection region (31.114) form the basis of

Student’st-test. It only remains to determine the sampling distributionP(t|H 0 ).

At the outset, it is worth noting that if we write the expression (31.113) fort

in terms of the standard estimatorσˆ=

√

Ns^2 /(N−1) of the standard deviation

then we obtain

t=

̄x−μ 0

σ/ˆ

√

N

. (31.115)

If, in fact, we knew the true value ofσanduseditinthisexpressionfortthen

it is clear from our discussion in section 31.3 thattwould follow a Gaussian

distribution with mean 0 and variance 1, i.e.t∼N(0,1). Whenσis not known,

however, we have to use our estimateσˆ in (31.115), with the result thattis

no longer distributed as the standard Gaussian. As one might expect from

the central limit theorem, however, the distribution oftdoes tend towards the

standard Gaussian for large values ofN.

As noted earlier, the exact distribution oft, valid for any value ofN,wasfirst

discovered by William Gossett. From (31.35), if the hypothesisH 0 is true then the

joint sampling distribution of ̄xandsis given by

P(x, s ̄ |H 0 )=CsN−^2 exp

(

−

Ns^2

2 σ^2

)

exp

[

−

N( ̄x−μ 0 )^2

2 σ^2

]

,

(31.116)

whereCis a normalisation constant. We can use this result to obtain the joint

sampling distribution ofsandtby demanding that

P( ̄x, s|H 0 )d ̄xds=P(t, s|H 0 )dt ds.